40 CHAPTER 2. LINEAR TRANSFORMATIONS

Here is another example.

Example 2.1.4 Compute  1 2 1 3

0 2 1 −2

2 1 4 1



1

2

0

1

 .

First of all, this is of the form (3× 4) (4× 1) and so the result should be a (3× 1) .Note how the inside numbers cancel. To get the entry in the second row and first and onlycolumn, compute

4∑k=1

a2kvk = a21v1 + a22v2 + a23v3 + a24v4

= 0× 1 + 2× 2 + 1× 0 + (−2)× 1 = 2.

You should do the rest of the problem and verify

 1 2 1 3

0 2 1 −2

2 1 4 1



1

2

0

1

 =

 8

2

5

 .

With this done, the next task is to multiply an m × n matrix times an n × p matrix.Before doing so, the following may be helpful.

(m× n) (n× p) = m× p

If the two middle numbers don’t match, you can’t multiply the matrices!

The number of columns on the left equals the number of rows on the right.

Definition 2.1.5 Let A be an m × n matrix and let B be an n × p matrix. Then B is ofthe form

B = (b1, · · · ,bp)

where bk is an n× 1 matrix. Then an m× p matrix AB is defined as follows:

AB ≡ (Ab1, · · · , Abp) (2.10)

where Abk is an m× 1 matrix. Hence AB as just defined is an m× p matrix. For example,

Example 2.1.6 Multiply the following.

(1 2 1

0 2 1

) 1 2 0

0 3 1

−2 1 1

The first thing you need to check before doing anything else is whether it is possible to

do the multiplication. The first matrix is a 2×3 and the second matrix is a 3×3. Therefore,